Dynamics of Microscale and Nanoscale Systems in the Weak-Memory Regime

Area of Science:

• Physics
• Statistical Mechanics
• Theoretical Physics

Background:

• Memory effects are common in small-scale systems.
• These effects arise from interactions between accessible and inaccessible degrees of freedom.
• Standard Markov approximation restores locality when timescales are sharply separated.

Purpose of the Study:

• To rigorously extend the Markov approximation to a weak-memory regime.
• To provide a unifying framework for describing memory effects in nonlocal time evolution equations.
• To derive error bounds and a convergent perturbation scheme for local approximations.

Main Methods:

• Extension of the standard Markov approximation.
• Derivation of explicit error bounds for local approximations.
• Development of a convergent perturbation scheme.

Main Results:

• The Markov approximation can be rigorously extended to a weak-memory regime.
• Explicit bounds on the error of local approximations are derived.
• A convergent perturbation scheme for constructing local approximations is presented.

Conclusions:

• The developed theory provides a unifying framework for describing memory effects.
• It is applicable to any autonomous and linear nonlocal time evolution equation.
• This work offers a systematic approach to handling memory effects in diverse physical systems.